The Hamiltonian for gravity reads [1]}, [2]}
\(H &=& \int d^3 x\, \left\lbrace \partial \bar{h}\, \bar{\partial }h\ +\ 2\, \kappa \, \partial _-^2 \bar{h}\, \left(h\,\frac{{\bar{\partial }}^2}{\partial _-^2}h\ -\ \frac{\bar{\partial }}{\partial _-}h\, \frac{\bar{\partial }}{\partial _-}h\right) + c.c.\ +\ \mathcal {O}(\kappa ^2)\right\rbrace \,.\) 
